# Thread: Concept of the Limit

1. ## Concept of the Limit

When I first learned about the limit it greatly disturbed my algebraic thinking, but I pushed the thought aside for a while, and now it's come back.

Take for example:

$\displaystyle \lim_{x \to 2}\frac{x^2-4}{x^2-2x}$

We have the expression

$\displaystyle \frac{x^2-4}{x^2-2x}$, which is equivalent to $\displaystyle \frac{x+2}{x}$

On the left, substituting in x we get an undefined answer, substituting into the right expression however, we get 2.

Aren't the two expression as perfect as each other? Then why do they give a different result when you substitute in x?
It's like we are, by manipulating the number, forcing it to give a certain answers against its will.

2. When $\displaystyle x\to 2$ that means that $\displaystyle x$ is in a neighbourhood of $\displaystyle 2$ and $\displaystyle x\neq 2$.
So the expression $\displaystyle \displaystyle\frac{x^2-4}{x^2-2x}$ can be simplify by $\displaystyle x-2$, because $\displaystyle x\neq 2$.

3. Originally Posted by DivideBy0
When I first learned about the limit it greatly disturbed my algebraic thinking, but I pushed the thought aside for a while, and now it's come back.

Take for example:

$\displaystyle \lim_{x \to 2}\frac{x^2-4}{x^2-2x}$

We have the expression

$\displaystyle \frac{x^2-4}{x^2-2x}$, which is equivalent to $\displaystyle \frac{x+2}{x}$
Only when $\displaystyle x \ne 2$, because to get the second you will have divided the first by zero (which is not allowed). So while this is true when $\displaystyle x \ne 2$ you cannot just substitute $\displaystyle 2$ into both sides and expect equality.

This is one of the reasons you are not allowed to divide by zero.

RonL